If a subset of a partially ordered set has exactly one minimal element, must that element be a smallest element?


Solution 1

Consider for instance $A := \mathbb{Z} \cup \{ a, b\}$, where we order the elements from $\mathbb{Z}$ in the usual way, and we define $a < b$, thus creating a partial order on $A$. Because there is no $x \in A$ such that $x < a$, $a$ is a minimal element, but not the smallest element because $a < x$ is not satisfied for all $x \in A$. After all, $a < x$ is not defined for any $x \in \mathbb{Z}$. Clearly there is no minimal element in $\mathbb{Z}$, so the condition is satisfied.

One doesn't have to use $\mathbb{Z}$ in particular, but what made it easy is the fact that it is an infinite set (in the sense that I couldn't think of any counterexamples for finite sets). The reason why I used $\mathbb{Z}$ is just because it comes with a default order according to which it has no minimal element (or smallest, for that matter).

Solution 2

It would be true in a finite set.

In fact, suppose $P$ is a partially ordered set with exactly one minimal element $x$, but $x$ is not a smallest element. Then the set $S$ of $s \in P$ which are not comparable to $x$ is nonempty. If $S$ has a minimal element $y$, it can't be a minimal element of $P$, so there is some $z \in P\backslash S$ with $z < y$. But then $x \le z < y$, contradiction. So $S$ has no minimal element, which implies there is an infinite descending chain $s_1 > s_2 > s_3 > \ldots$ in $S$.


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Computer Science Student at Calstate LA, motorcycle enthusiast and Rick and Morty megafan. Let's get rickity, rickity wrecked son!

Updated on August 01, 2022


  • Squanchinator
    Squanchinator over 1 year

    Give either a proof or a counter example to justify your answer.

    It's intuitive to think that if a partial order has exactly one minimal element,then that element must be the smallest element.

    However, the back of the book says otherwise.

    Can someone provide some counter examples to solidify the idea.

    • Juanito
      Juanito over 6 years
      How do you define minimal and smallest?
    • Blackbird
      Blackbird over 6 years
      Consider $X:=\mathbb{Z}\cup\{A,B\}$. Use the usual order on $\mathbb{Z}$ and declare $A$ smaller than $B$. Leave the parts $\mathbb{Z}$ and $\{A,B\}$ incomparable. Then, $A$ is the only minimal element but there is no smallest element.